Page 321 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

298 BIOMECHANICS OF THE HUMAN BODY

force-length relationship (Gordon et al., 1966), Fv( ) represents the force-velocity relationship
m
V

(Hill, 1938; Zajac, 1989; Epstein and Herzog, 1998), and Fl() represents the passive elastic force-
m
P
length relationship (Schutte, 1992), and b is the damping factor (Schutte et al., 1993). These parameters
m
m
are normalized to the maximum isometric muscle force, optimal fiber length (I ), and maximum
o
muscle contraction velocity (v ).
max
Pennation angle (ϕ(t)) changed with instantaneous muscle fiber length by assuming that muscle
has constant thickness and volume as it contracts (Scott and Winter, 1991). A typical equation to cal-
culate pennation angle, ϕ(t) is
l m sin ϕ
−1
=
t
ϕ() sin ( o o ) (12.7)
l m () t
m
t
where l () is the muscle fiber length at time t, and ϕ is the pennation angle at muscle optimal fiber
o
m
length l .
o
The contractile element’s force-length relationship Fl() is a curve created by a cubic spline
A
m
interpolation of the points on the force-length curve defined by Gordon et al. (1966). Huijing (1996)
has shown that optimal fiber lengths increase as activation decreases, an observation which has also
been reported by Guimaraes et al. (1994). This coupling relationship between muscle activation and
optimal fiber length can be incorporated into the muscle model (Lloyd and Besier, 2003) as
m
m
l () = l ( ( − a t()) +1 ) (12.8)
λ
t
1
o
o
m
where λ is the percentage change in optimal fiber length, and l () is the optimal fiber length at time t.
t
o

The passive elastic force-length function Fl() is given by an exponential relationship that is
m
P
described by Schutte (1992).

Fl() = e 10 ( l m −1 ) (12.9)
m
P
e 5

The force-velocity relationship Fv( ) and damping element can be used based on that employed
V
m
by Schutte et al. (1993). The damping element accounts for the intrinsic damping characteristics of
muscles, and improves the stability of the model.
Tendon force varies with tendon strain ε only when tendon length l t is greater than tendon slack
t
t
length l ; otherwise the tendon force is zero. Subsequently the normalized tendon force, F , is given
s
by (Zajac, 1989)
t
F = 0 ε ≤ 0
t
ε
.
F = 1480 3ε. 2 0 < < 0 0127 (12.10)
t
F = 37 5ε. − 0 2 . 3375 ε≥ 0 0127
.
where tendon strain is defined as
t
ε=(l t − ) l s t (12.11)
l
s
Muscle tendon length and activation data can be used as input to the muscle model, and then mus-
cle fiber lengths can then be calculated by forward integration of the fiber velocities obtained from
the equilibrium between tendon force and muscle fiber force using a Runge-Kutta-Fehlberg algo-
rithm. After the muscle fiber lengths and velocities are determined, the muscle force can be calcu-
lated through Eq. (12.6). Once individual muscle forces are estimated, they are multiplied by muscle
moment arms and summed to determine total joint moments.